A pipe heated from 20 °C to 170 °C grows by about 1.8 mm for every metre of its length. Held rigidly at both ends, a straight run cannot grow at all, and the stress needed to stop it is about 360 MPa in carbon steel, more than its yield strength. Real lines survive because they are not straight: every change of direction lets the pipe bend, and bending absorbs the growth at a small fraction of that stress. Piping flexibility analysis asks whether a layout is flexible enough: how far the line moves, what it does to the equipment and structure it is attached to, and what stresses its movement and its weight cause (Part II A).

Those questions belong to the line as a whole, and for them a piping system can be described by the line that runs down the middle of its pipe. Components, supports, anchors and loads are all attached to that centreline. SWBPIPE divides it into beam elements that meet at nodes, each node free to move in three directions and to turn about three axes, and solves one system of equations for the movements of all the nodes. From them it recovers the loads on every support and anchor, the forces and moments along every element, and the stresses those produce (Part III).

1.What the program computes, and what comes from you.

SWBPIPE ships with no code tables, and it does not judge compliance with any code. Its results are for working a design out and understanding it; the design is validated elsewhere (Part V).

A beam model knows nothing of what happens inside a pipe wall. A bend under moment flattens into an oval, which makes it several times more flexible than a straight pipe of the same length and concentrates stress at its flanks. A tee or a nozzle carries load through the local bending of shells. The centreline model takes these effects in through two numbers per component, both found by shell theory and by test: the flexibility factor k, which softens the element, and the stress intensification factor i, which raises the stress read from it (Part II E). Those two numbers are the bridge between beam and shell, and most of a flexibility analysis’s accuracy depends on them.

Where the bridge does not reach, the global model is not enough. Nozzle loads on a vessel or tank, trunnions, lugs, large branch connections and other places where load enters the pipe locally may need a local model: a method such as WRC 107 or 297, or finite elements. The global model supplies the boundary loads and movements for that model; it does not replace it.

A result is the answer to the model as entered, not a property of the plant. Reading one well means asking whether the centreline, supports and restraints match the line as built, whether every input is complete and right, and whether a local detail needs closer study. Part IV describes how to build and check a model.

Systematic study of pipe bends began in 1910, when Bantlin reported that curved pipes flatten under bending and are far more flexible than beam theory allows. A year later von Kármán published the first theory of it. That line of work, carried on by others and brought to closed form by Clark and Reissner in 1951, gave the flexibility factors engineers still use.

Analytical methods for whole lines came first for pipe lying in one plane. In 1929 W. H. Shipman published a general solution of the single-plane problem with two fixed ends, by integrating in advance the quantities that depend only on the shape of the line; the method needed only simple arithmetic and suited routine design. Tingey introduced the elastic centre, the centroid of the line’s elastic weight, about which the equations uncouple. Graphical constructions, charts and bench tests of scale models gave quicker answers for simple layouts and checked the calculations.

In 1932 the Engineering Department of the Standard Oil Company (Indiana) extended Shipman’s method to lines in space, in a company report. The M. W. Kellogg Company carried that work forward, organised it for its own use in the 1930s, and in 1941 published it as Design of Piping Systems: Expansion Stresses and Reactions in Piping Systems, written by D. B. Rossheim, A. R. C. Markl, H. V. Wallstrom and E. Slezak: the first general method for the stresses, reactions and movements of a piping system in space, with any number of legs, bends and restraints.

1.From stress to stress range.

In 1940 Rossheim and Markl argued that thermal stress is self-limiting, so a line fails not by one overload but by fatigue, and that it should be judged by the range of stress through its cycles. They proposed an allowable range and began the fatigue tests of piping components that Markl carried through the 1940s and early 1950s, from which the stress intensification factors come. Markl’s “Piping-Flexibility Analysis” of 1955 set out the approach that the Code for Pressure Piping, ASA B31.1, adopted in its 1955 edition, and that the B31 codes still follow (Part II D).

The second edition of Design of Piping Systems (John Wiley & Sons, 1956) widened the book to the whole design of piping systems and brought in the new stress-range rules. It is the edition most stress engineers know.

2.The computer.

The general method was built of tabulated steps, and it passed naturally to the computing machine. Kellogg ran routine piping solutions by machine from 1954. The matrix methods of structural analysis followed, and in the decades after them the proprietary pipe stress programs used today. SWBPIPE belongs to that later family: it solves the same problem by the stiffness method, and its bend element is built from the same flexibility integrals the Kellogg method used. What it adds is that every step can be read.

The 1941 manual opens with an acknowledgment. Among the many investigators whose work it built on, it singles out three. It pays particular tribute to W. H. Shipman, whose systematic solution of the single-plane problem, by preintegration of the shape coefficients, paved the way for what followed. And it credits G. W. Watts and W. R. Burrows, of the Engineering Department of the Standard Oil Company (Indiana), with

“the most significant step towards the general solution” Design of Piping Systems, 1941, Acknowledgment

by extending Shipman’s method to the three-plane problem with two fixed ends. Their work, published only as a company report, derived the coefficients for the other planes from those of one basic plane by a cyclic permutation of the coordinates. These works, the acknowledgment says, underlie the general method the manual presents.

SWBPIPE takes its name from the three of them. For decades the general method was worked by hand, one coefficient at a time; it has since moved into programs where it is out of sight. SWBPIPE is made to keep it in the open: the mechanics, the evidence and the source, within reach of anyone willing to learn them. The expansion loop in Part VI is worked by the force method the 1941 manual made general, and checked against the program.